In my first column I suggested that many children experience mathematics as no more than the memorisation of facts, rules, formulas and procedures needed to determine the answers to questions. I also said that because, in most cases, the rules make little sense, these children develop maths anxiety, partly because they feel powerless when it comes to mathematics. By contrast, children who experience mathematics as a meaningful, sense-making and problem-solving activity are less likely to experience these anxieties.
Having discussed the importance of a strong sense of number and ways of helping children to develop this, let’s look at the role of problems in developing mathematical confidence and competence.
Problems are often thought of as a reason for studying mathematics. Mathematicians solve problems and, to do this, they need to be able to “do” mathematics.
What is not as well appreciated is that problems also provide a way of introducing children to mathematics: learning the subject by solving problems. This is true for mathematics at all levels; it is particularly so for children in the early years.
Consider for a moment a young child, say four years old. The child’s mother gives her some sweets with the instruction to share them with her little brother. Chances are good that she will be able to make a plan that will result in the equal sharing of the sweets with her brother. Her strategy may well involve a ‘one for you, one for me, one for you, one for me …” in dealing out of the sweets.
In mathematical terms the girl has divided the sweets by two and the number of sweets that she and her brother each get can be determined by dividing the number of sweets by two. In mathematical terms the girl has completed a division problem and yet she may well be unable to count and may never have heard the word “division”.
Young children have a natural ability to solve problems that can be used to good effect in the introduction of mathematics. Not only does this offer a way of introducing mathematics and, in particular, the four basic operations (addition, subtraction, multiplication and division) of the foundation phase, but it also helps them see value in the work they do, thus gaining confidence that is important in helping them to succeed in mathematics.
Here’s a problem I have used to good effect with children in a foundation phase class. I posed the following problem: The tuckshop has made 27 amagwinya (vetkoek). There are 43 children in the class. Are there enough amagwinya for each child to get one?
After an enthusiastic classroom discussion, it was agreed that there were not enough amagwinya. I asked the children to determine how many more were needed.
Odwa solved the problem by first drawing 27 stripes to represent the 27 children in the class and then he drew a large number of extra stripes. He then counted on from 27: 28, 29, 30, and so on, to 43 and highlighted the 43rd stripe. Finally, he counted how many extra amagwinya (stripes) were needed and concluded that 16 more amagwinya needed to be made.
Mathematically, we can summarise Odwa’s solution as follows: 27 + 16 = 43. That is, we can think of Odwa as having added on from 27 to 43 and, in so doing, he established that 16 additional amagwinya were needed.
Asavelo solved the problem by first counting out 43 counters. Next she counted out 27 from the 43 — as if she were giving amagwinya to those to whom she could give. Finally, she counted the remaining counters and established that she still needed 16 amagwinya for the remaining children.
Mathematically, we can summarise Asavelo’s solution as follows: 43-27=16. That is, we can see that Asavelo subtracted 27 from 43 to establish that 16 additional amagwinya were needed.
These two examples illustrated a problem that could, in mathematical terms, be solved by means of either addition or subtraction.
After a number of similar problems and appropriate discussion of both Odwa’s and Asavelo’s approaches, the teacher’s role becomes one of introducing the mathematical vocabulary and notation associated with these perfectly natural problem-solving strategies.
My point is that, as a teacher, I can choose either to introduce a lesson (in the foundation phase) on addition and the rules and methods associated with the procedure, or I can present children with a problem that will provoke them into performing an action that is referred to as addition.
After that, it remains for me to introduce the vocabulary, symbols and conventions associated with the perfectly natural procedure.
There is one more observation I want to share here. Notice that both Odwa and Asavelo were confident about their answers. They could check them against the context. They had accounted for 43 children, they had used up the 27 amagwinya available and they could justify the need for an additional 16. For many children, the only way they have of knowing whether or not their answer to the sums: 43-27= . . . and 27+ . . . = 43 is correct is to ask their teacher.
As Odwa’s and Asavelo’s number sense improves, so they will rely less on stripes and counters and become able to use numbers with greater confidence in solving the same problems.
I believe strongly that children who learn mathematics through problems will see the value of and be able to make sense of what they are doing far more easily than those who don’t.
In my next column I will take the discussion of problems and the teacher’s role in using them to good effect further.
Aarnout Brombacher is a private mathematics consultant